Background: Since the introduction of the SARS-CoV-2 vaccine, the incidence of complications associated with pediatric infection has markedly declined. However, there is a notable decline in vaccination coverage among younger age groups. The objective of this study was to identify the factors associated with hesitancy to receive the SARS-CoV-2 vaccine in parents of children aged 3–15 years in a Latin American context. Methods: A cross-sectional, analytical study was conducted between February and May 2023. Parents of children between the ages of 3 and 15, residing in the metropolitan region of Chile, were surveyed via an online or in-person questionnaire. Univariate analysis and a multivariate logistic regression model were performed. Results: Five hundred thirty-nine surveys were completed; 61.6% of parents were aged between 21 and 40 years, 42.3% had completed higher education, 79.4% had paid employment, and 98.1% were fully vaccinated. Age distribution of children was 36.9% for 3–5 years, 42.4% for 6–11 years, and 20.8% for 12–15 years. 87.4% was vaccinated. In the univariate analysis, vaccine hesitancy was observed in 9%, associated mainly with children 3–5 years, fewer doses in parents’ vaccination schedule, lacking risk perception and concerns about safety, and not knowing where to access pediatric vaccine formulations against COVID-19. In the multivariate analysis, the younger age, lacking risk perception, and concerns about the vaccine’s effectiveness were associated with vaccine hesitancy. Conclusion: Addressing hesitancy toward COVID-19 vaccines is of utmost importance. Educational interventions should be implemented to inform parents about the importance and effectiveness of COVID-19 vaccination in children, with special emphasis on parents of younger children, to increase vaccination coverage.